How do evolutionary algorithms have advantages over the conventional backpropagation methods?
Unlike backpropagation, evolutionary algorithms do not require the objective function to be differential with respect to the parameters you aim to optimize. As a result, you can optimize "more things" in the network, such as activation functions or number of layers, which wouldn't be possible in the standard backpropagation. 

Another advantage is that by defining the mutation and crossover functions, you can influence how the parameter search space should be explored.